Supplemental Information C: Power Simulations

Supplemental Material for ‘Need Fulfillment During Intergroup Contact: Three Experience Sampling Studies’

Note. Boxplots display the interquartile range (IQR, center box), and the whiskers extend 1.5*IQR from the lower and upper hinge. The white point indicates the mean and the white center line indicates the median.

Generally, power considerations of mixed effects model such as repeated measures ANCOVAs and multi-level regressions are difficult to estimate because of the complex covariance structures. We ran simulation-studies based on our initial results following the first study. We found that sufficient sample sizes heavily depended on the type and size of the effect. When we simulated models with the marginalization sub-scale (which was the best distributed with mostly small to medium effect sizes), we found that for between-subjects effects, a sample size of 80 participants would have been sufficient (power above .8) with two measurements (see Figure 9 for an illustration). However, for the same between-subject effects we did not reach a power above .8 with any increase of repeated follow-up measurements (based on model extrapolations with a fixed n = 20; tested up to 12 months). The within time effect, on the other hand, was estimated well above .8 by the seventh measurement (i.e., after six months, see Figure 9). The cross-level interaction effects (within-between) were largely a mixture of their individual effects and changes in measurement numbers and sample size affected the models accordingly. We, therefore, urge researchers to rely on pilot data and to run extensive simulation studies to maximize effort and feasibility (i.e., sufficient power).

Data Import

load("data/S1_Workers/processed/df.btw.RData")

Extract Effects for Simulation

Prepare Data

# prepare data for regression (i.e, put into long format and change variable types)
marginalization = melt(df.btw[!is.na(df.btw$assimilation.post),c("marginalization.pre","marginalization.post",
                                                                 "CtContactNL_c","AvKeyNeedInt_c",
                                                                 "ExternalReference")],
                       id=c("ExternalReference","CtContactNL_c","AvKeyNeedInt_c"))
marginalization$time = ifelse(grepl(".pre", marginalization$variable), "pre", "post")
marginalization$variable = gsub(".pre$|.post$", "", marginalization$variable)
marginalization$time = factor(marginalization$time)
marginalization$time = factor(marginalization$time,levels(marginalization$time)[c(2,1)])
marginalization$id = rep(seq(1:20), 2)

ML Regression

Run multilevel regression to extract the effect sizes and covariance matrices

# run regresion 
lm.marg <- lmer(value~
                  AvKeyNeedInt_c*time+
                  CtContactNL_c*time+
                  (1|id), 
                data=marginalization)
summary(lm.marg)

## Linear mixed model fit by REML. t-tests use Satterthwaite's method ['lmerModLmerTest']
## Formula: value ~ AvKeyNeedInt_c * time + CtContactNL_c * time + (1 | id)
##    Data: marginalization
## 
## REML criterion at convergence: 167
## 
## Scaled residuals: 
##    Min     1Q Median     3Q    Max 
## -1.743 -0.375 -0.004  0.421  1.428 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  id       (Intercept) 3.268    1.808   
##  Residual             0.934    0.966   
## Number of obs: 40, groups:  id, 20
## 
## Fixed effects:
##                         Estimate Std. Error      df t value   Pr(>|t|)    
## (Intercept)               3.2245     0.4690 21.1848    6.87 0.00000082 ***
## AvKeyNeedInt_c           -0.0606     0.0395 21.1848   -1.53      0.140    
## timepost                 -0.4957     0.3127 17.0000   -1.59      0.131    
## CtContactNL_c            -0.0576     0.0415 21.1848   -1.39      0.180    
## AvKeyNeedInt_c:timepost   0.0615     0.0263 17.0000    2.34      0.032 *  
## timepost:CtContactNL_c    0.0208     0.0277 17.0000    0.75      0.462    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) AvKNI_ timpst CtCNL_ AKNI_:
## AvKyNdInt_c  0.107                            
## timepost    -0.333 -0.036                     
## CtCntctNL_c -0.202 -0.214  0.067              
## AvKyNdInt_: -0.036 -0.333  0.107  0.071       
## tmpst:CCNL_  0.067  0.071 -0.202 -0.333 -0.214

fixef(lm.marg)

##             (Intercept)          AvKeyNeedInt_c                timepost           CtContactNL_c AvKeyNeedInt_c:timepost  timepost:CtContactNL_c 
##                 3.22445                -0.06059                -0.49572                -0.05756                 0.06148                 0.02082

Post-Hoc Power

Check post-hoc power for comparison.

# Post-hoc power analysis for single effects (example: key need)
post_hoc <- powerSim(lm.marg, fixed("AvKeyNeed"), nsim=1000)

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post_hoc

## Power [user defined], (95% confidence interval):
##        0.00% ( 0.00,  0.37)
## 
## Test: [user defined function]
## 
## Based on 1000 simulations, (0 warnings, 1000 errors)
## alpha = 0.05, nrow = 40
## 
## Time elapsed: 0 h 1 m 23 s
## 
## nb: result might be an observed power calculation

Power Simulation Participant Numbers

Simulate different participant numbers for all fixed effects.

Power of contact effect for different numbers of participants

# extrapolate data:
sim.N <- extend(lm.marg, along="id", n=500)

# Run and plot simulations
plt.pwr.contact = powerCurve(fit = sim.N, 
                             test = simr::fixed("CtContactNL_c", "t"), 
                             along="id", 
                             progress = FALSE,
                             breaks = seq(10,100,10), 
                             nsim=1000)
plot(plt.pwr.contact) +
  title("Outgroup Contact Count [participant numbers]")

## integer(0)

Power of time effect for different numbers of participants

plt.pwr.time = powerCurve(sim.N, simr::fixed("time"), 
                          along="id", 
                          progress = FALSE, 
                          breaks = seq(10,100,10), 
                          nsim=1000)
plot(plt.pwr.time) +
  title("Time [participant numbers]")

## integer(0)

Power of need fulfillment effect for different numbers of participants

plt.pwr.Need = powerCurve(sim.N, simr::fixed("AvKeyNeedInt_c", "t"), 
                          along="id", 
                          progress = FALSE, 
                          breaks = seq(10,100,10), 
                          nsim=1000)
plot(plt.pwr.Need) +
  title("Avergage Need Fulfillment [participant numbers]")

## integer(0)

Power of time by contact effect for different numbers of participants

plt.pwr.timeCont = powerCurve(sim.N, simr::fixed("time:CtContactNL_c"), 
                             along="id", 
                             progress = FALSE, 
                             breaks = seq(10,500,50), 
                             nsim=1000)
plot(plt.pwr.timeCont) +
  title("Time X Outgroup Contact Count [participant numbers]")

## integer(0)

Power of time by need fulfillment effect for different numbers of participants

plt.pwr.timeNeed = powerCurve(sim.N, simr::fixed("AvKeyNeedInt_c:time"), 
                              along="id", 
                              progress = FALSE, 
                              breaks = seq(10,50,5), 
                              nsim=1000)
plot(plt.pwr.timeNeed) +
  title("Time X Avergage Need Fulfillment [participant numbers]")

## integer(0)

Power Simulation Measurement Numbers

Simulate different number of within person measurements.

Power of time effect for different numbers of measurements

# extrapolate data:
sim.within <- extend(lm.marg, within="id", n=12)

# Run and plot simulations
plt.pwr.t.time = powerCurve(sim.within, simr::fixed("time"), 
                          within="id", 
                          progress = FALSE, 
                          breaks = seq(2,12,1), 
                          nsim=1000)
plot(plt.pwr.t.time) +
  title("Time [measurement numbers]")

## integer(0)

Power of contact effect for different numbers of measurements

plt.pwr.t.contact = powerCurve(sim.within, simr::fixed("CtContactNL_c", "t"), 
                               within="id", 
                               progress = FALSE, 
                               breaks = seq(2,12,1), 
                               nsim=1000)
plot(plt.pwr.t.contact) +
  title("Outgroup Contact Count [measurement numbers]")

## integer(0)

Power of need fulfillment effect for different numbers of measurements

plt.pwr.t.Need = powerCurve(sim.within, simr::fixed("AvKeyNeedInt_c", "t"), 
                            within="id", 
                            progress = FALSE, 
                            breaks = seq(2,12,1), 
                            nsim=1000)
plot(plt.pwr.t.Need) +
  title("Average Need Fulfillment [measurement numbers]")

## integer(0)

Power of time by contact effect for different numbers of measurements

plt.pwr.t.timeCont = powerCurve(sim.within, simr::fixed("time:CtContactNL_c"), 
                          within="id", 
                          progress = FALSE, 
                          breaks = seq(2,12,1), 
                          nsim=1000)
plot(plt.pwr.t.timeCont) +
  title("Time X Outgroup Contact Count [measurement numbers]")

## integer(0)

Power of time by need fulfillment effect for different numbers of measurements

plt.pwr.t.timeNeed = powerCurve(sim.within, simr::fixed("AvKeyNeedInt_c:time"), 
                               within="id", 
                               progress = FALSE, 
                               breaks = seq(2,12,1), 
                               nsim=1000)
plot(plt.pwr.t.timeNeed) +
  title("Time X Average Need Fulfillment [measurement numbers]")

## integer(0)

Export data and plots

# save simulations, because this takes for freakin ever.      
save(plt.pwr.contact,plt.pwr.Need,plt.pwr.t.contact,plt.pwr.t.Need,
     plt.pwr.t.time,plt.pwr.t.timeCont,plt.pwr.t.timeNeed,plt.pwr.time,
     plt.pwr.timeCont,plt.pwr.timeNeed, 
     file = "data/S1_Workers/processed/PowerSimPlots.RData")

#load("PowerSimPlots.RData")

Software Information

The full session information with all relevant system information and all loaded and installed packages is available in the collapsible section below.

System Info

Table 1: R environment session info for reproducibility of results

Setting	Value
version	R version 4.3.0 (2023-04-21)
os	macOS Ventura 13.3.1
system	x86_64, darwin20
ui	X11
language	(EN)
collate	en_US.UTF-8
ctype	en_US.UTF-8
tz	Europe/Amsterdam
date	2023-05-19
pandoc	3.1.1 @ /usr/local/bin/ (via rmarkdown)

Package Info

Table 2: Package info for reproducibility of results

Package	Loaded version	Date	Source
binom	1.1-1.1	2022-05-02	CRAN (R 4.3.0)
bookdown	0.34	2023-05-09	CRAN (R 4.3.0)
brms	2.19.0	2023-03-14	CRAN (R 4.3.0)
car	3.1-2	2023-03-30	CRAN (R 4.3.0)
carData	3.0-5	2022-01-06	CRAN (R 4.3.0)
data.table	1.14.8	2023-02-17	CRAN (R 4.3.0)
devtools	2.4.5	2022-10-11	CRAN (R 4.3.0)
dplyr	1.1.2	2023-04-20	CRAN (R 4.3.0)
ggpattern	1.0.1	2022-11-09	CRAN (R 4.3.0)
ggplot2	3.4.2	2023-04-03	CRAN (R 4.3.0)
ggridges	0.5.4	2022-09-26	CRAN (R 4.3.0)
ggthemes	4.2.4	2021-01-20	CRAN (R 4.3.0)
gridExtra	2.3	2017-09-09	CRAN (R 4.3.0)
haven	2.5.2	2023-02-28	CRAN (R 4.3.0)
Hmisc	5.1-0	2023-05-08	CRAN (R 4.3.0)
iterators	1.0.14	2022-02-05	CRAN (R 4.3.0)
kableExtra	1.3.4	2021-02-20	CRAN (R 4.3.0)
knitr	1.42	2023-01-25	CRAN (R 4.3.0)
lme4	1.1-33	2023-04-25	CRAN (R 4.3.0)
lmerTest	3.1-3	2020-10-23	CRAN (R 4.3.0)
lubridate	1.9.2	2023-02-10	CRAN (R 4.3.0)
Matrix	1.5-4	2023-04-04	CRAN (R 4.3.0)
pander	0.6.5	2022-03-18	CRAN (R 4.3.0)
papaja	0.1.1	2022-07-05	CRAN (R 4.3.0)
pbkrtest	0.5.2	2023-01-19	CRAN (R 4.3.0)
plotly	4.10.1	2022-11-07	CRAN (R 4.3.0)
plotrix	3.8-2	2021-09-08	CRAN (R 4.3.0)
plyr	1.8.8	2022-11-11	CRAN (R 4.3.0)
psych	2.3.3	2023-03-18	CRAN (R 4.3.0)
purrr	1.0.1	2023-01-10	CRAN (R 4.3.0)
RColorBrewer	1.1-3	2022-04-03	CRAN (R 4.3.0)
Rcpp	1.0.10	2023-01-22	CRAN (R 4.3.0)
remedy	0.1.0	2018-12-03	CRAN (R 4.3.0)
reshape2	1.4.4	2020-04-09	CRAN (R 4.3.0)
RLRsim	3.1-8	2022-03-16	CRAN (R 4.3.0)
rmarkdown	2.21	2023-03-26	CRAN (R 4.3.0)
sessioninfo	1.2.2	2021-12-06	CRAN (R 4.3.0)
simr	1.0.7	2023-04-13	CRAN (R 4.3.0)
stringi	1.7.12	2023-01-11	CRAN (R 4.3.0)
stringr	1.5.0	2022-12-02	CRAN (R 4.3.0)
testthat	3.1.8	2023-05-04	CRAN (R 4.3.0)
tibble	3.2.1	2023-03-20	CRAN (R 4.3.0)
tinylabels	0.2.3	2022-02-06	CRAN (R 4.3.0)
usethis	2.1.6	2022-05-25	CRAN (R 4.3.0)

Full Session Info (including loaded but unattached packages — for troubleshooting only)

R version 4.3.0 (2023-04-21)

Platform: x86_64-apple-darwin20 (64-bit)

locale: en_US.UTF-8||en_US.UTF-8||en_US.UTF-8||C||en_US.UTF-8||en_US.UTF-8

attached base packages:

• grid
• stats
• graphics
• grDevices
• datasets
• utils
• methods
• base

other attached packages:

• purrr(v.1.0.1)
• lubridate(v.1.9.2)
• reshape2(v.1.4.4)
• stringi(v.1.7.12)
• papaja(v.0.1.1)
• tinylabels(v.0.2.3)
• Hmisc(v.5.1-0)
• devtools(v.2.4.5)
• usethis(v.2.1.6)
• tibble(v.3.2.1)
• plotly(v.4.10.1)
• haven(v.2.5.2)
• stringr(v.1.5.0)
• kableExtra(v.1.3.4)
• dplyr(v.1.1.2)
• plyr(v.1.8.8)
• pander(v.0.6.5)
• sessioninfo(v.1.2.2)
• lmerTest(v.3.1-3)
• data.table(v.1.14.8)
• simr(v.1.0.7)
• testthat(v.3.1.8)
• car(v.3.1-2)
• carData(v.3.0-5)
• RLRsim(v.3.1-8)
• plotrix(v.3.8-2)
• pbkrtest(v.0.5.2)
• lme4(v.1.1-33)
• Matrix(v.1.5-4)
• iterators(v.1.0.14)
• binom(v.1.1-1.1)
• ggridges(v.0.5.4)
• ggpattern(v.1.0.1)
• gridExtra(v.2.3)
• RColorBrewer(v.1.1-3)
• ggthemes(v.4.2.4)
• ggplot2(v.3.4.2)
• psych(v.2.3.3)
• brms(v.2.19.0)
• Rcpp(v.1.0.10)
• bookdown(v.0.34)
• remedy(v.0.1.0)
• knitr(v.1.42)
• rmarkdown(v.2.21)

loaded via a namespace (and not attached):

• shinythemes(v.1.2.0)
• splines(v.4.3.0)
• later(v.1.3.1)
• datawizard(v.0.7.1)
• rpart(v.4.1.19)
• xts(v.0.13.1)
• lifecycle(v.1.0.3)
• rprojroot(v.2.0.3)
• StanHeaders(v.2.21.0-7)
• processx(v.3.8.1)
• lattice(v.0.21-8)
• MASS(v.7.3-58.4)
• insight(v.0.19.1)
• crosstalk(v.1.2.0)
• backports(v.1.4.1)
• magrittr(v.2.0.3)
• sass(v.0.4.6)
• remotes(v.2.4.2)
• jquerylib(v.0.1.4)
• yaml(v.2.3.7)
• httpuv(v.1.6.11)
• pkgbuild(v.1.4.0)
• minqa(v.1.2.5)
• pkgload(v.1.3.2)
• abind(v.1.4-5)
• rvest(v.1.0.3)
• nnet(v.7.3-18)
• tensorA(v.0.36.2)
• inline(v.0.3.19)
• bridgesampling(v.1.1-2)
• svglite(v.2.1.1)
• codetools(v.0.2-19)
• DT(v.0.27)
• xml2(v.1.3.4)
• tidyselect(v.1.2.0)
• bayesplot(v.1.10.0)
• farver(v.2.1.1)
• effectsize(v.0.8.3)
• matrixStats(v.0.63.0)
• stats4(v.4.3.0)
• base64enc(v.0.1-3)
• webshot(v.0.5.4)
• jsonlite(v.1.8.4)
• Formula(v.1.2-5)
• ellipsis(v.0.3.2)
• emmeans(v.1.8.6)
• systemfonts(v.1.0.4)
• tools(v.4.3.0)
• glue(v.1.6.2)
• mnormt(v.2.1.1)
• xfun(v.0.39)
• mgcv(v.1.8-42)
• distributional(v.0.3.2)
• loo(v.2.6.0)
• withr(v.2.5.0)
• numDeriv(v.2016.8-1.1)
• fastmap(v.1.1.1)
• boot(v.1.3-28.1)
• fansi(v.1.0.4)
• shinyjs(v.2.1.0)
• callr(v.3.7.3)
• digest(v.0.6.31)
• timechange(v.0.2.0)
• R6(v.2.5.1)
• mime(v.0.12)
• estimability(v.1.4.1)
• colorspace(v.2.1-0)
• gtools(v.3.9.4)
• markdown(v.1.6)
• threejs(v.0.3.3)
• utf8(v.1.2.3)
• tidyr(v.1.3.0)
• generics(v.0.1.3)
• renv(v.0.14.0)
• prettyunits(v.1.1.1)
• httr(v.1.4.6)
• htmlwidgets(v.1.6.2)
• parameters(v.0.21.0)
• pkgconfig(v.2.0.3)
• dygraphs(v.1.1.1.6)
• gtable(v.0.3.3)
• brio(v.1.1.3)
• htmltools(v.0.5.5)
• profvis(v.0.3.8)
• scales(v.1.2.1)
• posterior(v.1.4.1)
• rstudioapi(v.0.14)
• coda(v.0.19-4)
• checkmate(v.2.2.0)
• nlme(v.3.1-162)
• nloptr(v.2.0.3)
• cachem(v.1.0.8)
• zoo(v.1.8-12)
• parallel(v.4.3.0)
• miniUI(v.0.1.1.1)
• foreign(v.0.8-84)
• pillar(v.1.9.0)
• vctrs(v.0.6.2)
• shinystan(v.2.6.0)
• urlchecker(v.1.0.1)
• promises(v.1.2.0.1)
• cluster(v.2.1.4)
• xtable(v.1.8-4)
• htmlTable(v.2.4.1)
• evaluate(v.0.21)
• mvtnorm(v.1.1-3)
• cli(v.3.6.1)
• compiler(v.4.3.0)
• rlang(v.1.1.1)
• crayon(v.1.5.2)
• rstantools(v.2.3.1)
• rematch2(v.2.1.2)
• ps(v.1.7.5)
• forcats(v.1.0.0)
• fs(v.1.6.2)
• rstan(v.2.21.8)
• viridisLite(v.0.4.2)
• munsell(v.0.5.0)
• lazyeval(v.0.2.2)
• colourpicker(v.1.2.0)
• bayestestR(v.0.13.1)
• Brobdingnag(v.1.2-9)
• hms(v.1.1.3)
• shiny(v.1.7.4)
• highr(v.0.10)
• fontawesome(v.0.5.1)
• memoise(v.2.0.1)
• igraph(v.1.4.2)
• broom(v.1.0.4)
• RcppParallel(v.5.1.7)
• bslib(v.0.4.2)

---